Index painting refers to creating Pixel Art in Indexed Modes, working with Indexed Images while paying attention to Palette Structure and keeping possible Palette Effects in mind and at all times keeping control over (if necessary by fixing the results) the pixel level polish of a piece even when using tools that influence more than one pixel at a time.
Historically, “Index Painting” is the natural way to go and think about making Pixel Art as most hardware and digital image editing tools either used Indexed Modes as the default mode or simply did not provide any other modes. The term “Index Painting” was not around then because there simply was nothing else. Today the term seems to be used to set “Pixel Art”(in the sense of Index Painting) apart from other “Pixel Art”(in the sense of Digital Painting and Computer Generated Imagery(e.g. 3D renderings)) and the third kind of “Pixel Art”(the purists “one pixel at a time”- process).
Some advocate placing one pixel at a time as the only “allowed” process to create “Pixel Art”(a term which eludes definition itself) while others advocate whether something is or is not “Pixel Art” can not be defined by processes but only by looking at the results.
The actual term “Index Painting” emerged from the on-going online debate about “pixel purism”, processes and results.